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Alternating series test

  1. Nov 29, 2008 #1
    1. The problem statement, all variables and given/known data

    Find a sequence [tex]a_{n}[/tex] which is non-negative and null but where [tex]\sum (-1)^{n+1} a_{n}[/tex] is divergent.

    2. Relevant equations

    Alternating series test:

    Let [tex]a_{n}[/tex] be a decreasing sequence of positive real numbers such that [tex]a_{n}\rightarrowa[/tex] as [tex]n\rightarrow\infty[/tex]. Then the series [tex]\sum (-1)^{n+1} a_{n}[/tex] converges.

    3. The attempt at a solution

    I'm a bit confused by this one. If [tex]a_{n}[/tex] is non-negative and null then it seems like it's decreasing to zero, in which case it satisfies the alternating series test. So how can the sum diverge?!
     
    Last edited: Nov 29, 2008
  2. jcsd
  3. Nov 29, 2008 #2

    Dick

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    How about (1,0,1/2,0,1/3,0,1/4,0...)? It's null but nondecreasing. The (-1)^(n+1) doesn't help much does it?
     
  4. Dec 1, 2008 #3
    Ah yeah i see. So you sort of pad it out with zeros to remove the minus terms. Thanks!
     
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